MRB=$\sum _{x=0}^{\infty } e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right) $ vs M2= $\int_0^\infty e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right) dx$ in proper integrals
See this notebook. I got an interesting co-answer here.
So, I could conclude that
The following "proof of it is from combining
https://www.quora.com/Why-is-it-that-f-x-e-i-pi-x-left-1-x-1-frac-1-x-1-right-int0-infty-f-t-dt-i-int0-infty-f-i-t-dt
and
https://www.quora.com/How-would-you-prove-for-g-x-x-1-x-lim-limits-N-to-infty-int-limits1-2N-e-i-pi-t-g-t-dt-i-lim-limits-N-to-infty-int-limits-0-2N-frac-g-1-it-exp-pi-t-dt.
While
Something similar is shown:
That looks like it proves to me. But I don't see any proof for .
I won't bring the subject of numerical computation into this discussion, but I spent several years learning to compute the digits of $\int_0^\infty{e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right)}dx.$ You can read about my adventure at How to calculate the digits of the MKB constant. If you like numeric computations, of much interest is the story of how I came across this integral, by investigating $\sum _{x=0}^{\infty } e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right)$ since the 1990s, at Try to beat these MRB constant records!
